The "official" MAME program can be found at mamedev.org. MAME is opensource, thus people can create their own builds of MAME- some with it's own added features. Often these custom builds are referred to as "a different flavor of MAME" or, with popular flavors, by the chosen name such as "MAMEUI".
The loose term "flavor" is chosen over "version". It is common to keep the version in sync to the MAME version it was derived from. This to avoid confusion as to what set of ROMs work with the modified MAME version. Note that software versioning might still be in place using build numbers.
Compiling your own MAME
An excellent guide can be found here (written by Mr.Do!) BYOAC forum user MKChamp has a guide and the files needed to create your own flavor of MAME without the so-called nag screen and support for highscores. The thread can be found here.
A list of popular MAME flavors:
- MAME - The command line program that the other variants are based on. 
- NoNag/hiscore patched - Suppresses error messages that are useful when troubleshooting. See the BYOAC forum thread .
- MAME32, MAMEUI, or MAMEUIFX - Graphic User Interface (GUI) versions of MAME. 
- HBMAME - Dedicated to homebrew "hacks" 
- GroovyMAME